Characterizing binary matroids with no P 9 -minor

1 2 Characterizing binary matroids with no P 9 -minor Guoli Ding 1 and Haidong Wu 2 1. Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA Email: ding@math.lsu.edu 2. Department of Mathematics, University of Mississippi, University, Mississippi, USA Email: hwu@olemiss.edu 3 4 5 6 7 8 9 10 11 12 Abstract In this paper, we give a complete characterization of binary matroids with no P 9 -minor. A 3-connected binary matroid M has no P 9 -minor if and only if M is one of the internally 4-connected non-regular minors of a special 16-element matroid Y 16, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid M (K 3,n ), M (K 3,n), M (K 3,n), or M (K 3,n) (n 2). Here the simple graphs K 3,n, K 3,n, and K 3,n are obtained from K 3,n by adding one, two, or three edges in the color class of size three, respectively. 13 1 Introduction 16 14 It is well known that the class of binary matroids consists of all matroids 15 without any U 2,4 -minor, and the class of regular matroids consists of matroids without any U 2,4, F 7 or F7 -minor. Kuratowski s Theorem states that a graph 17 is planar if and only if it has no minor that is isomorphic to K 3,3 or K 5. These 18 examples show that characterizing a class of graphs and matroids without 19 certain minors is often of fundamental importance. We say that a matroid is 20 N-free if it does not contain a minor that is isomorphic to N. A 3-connected 21 matroid M is said to be internally 4-connected if for any 3-separation of M, 22 one side of the separation is either a triangle or a triad. 23 24 25 26 27 28 29 There is much interest in characterizing binary matroids without small 3-connected minors. Since non-3-connected matroids can be constructed by 3-connected matroids using 1-, 2-sum operations, one needs only determine the 3-connected members of a minor closed class. There is exactly one 3- connected binary matroid with 6-elements, namely, W 3 where W n denotes both the wheel graph with n-spokes and the cycle matroid of W n. There are exactly two 7-element binary 3-connected matroids, F 7 and F7. There are 1

Figure 1: A geometric representation of P 9 30 31 32 33 three 8-element binary 3-connected matroids, W 4, S 8 and AG(3, 2), and there are eight 9-element 3-connected binary matroids: M(K 3,3 ), M (K 3,3 ), Prism, M(K 5 \e), P 9, P9, binary spike Z 4 and its dual Z4. E(M) Binary 3-connected matroids 6 W 3 7 F 7, F7 8 W 4, S 8, AG(3, 2) 9 M(K 3,3 ), M (K 3,3 ), M(K 5 \e), P rism, P 9, P9, Z 4, Z4 40 34 For each matroid N in the above list with less than nine elements, with 35 the exception of AG(3, 2), the problem of characterizing 3-connected binary 36 matroids with no N-minor has been solved. Since every 3-connected binary 37 matroid having at least four elements has a W 3 -minor, the class of 3-connected 38 binary matroids excluding W 3 contains only the trivial 3-connected matroids 39 with at most three elements. Seymour in [11] determined all 3-connected binary matroids with no F 7 -minor (F7 -minor). Any such matroid is either 41 regular or is isomorphic to F7 7). In [8], Oxley characterized all 3-connected 42 binary W 4 -free matroids. These are exactly M(K 4 ), F 7, F7, binary spikes Z r, 43 Zr, Z r \t, or Z r \y r (r 4) plus the trivial 3-connected matroids with at most three elements. It is well known that F 7, F7, and AG(3, 2) are the only 45 3-connected binary non-regular matroids without any S 8 -minor. 46 47 48 In the book [3], Mayhew, Royle and Whittle characterized all internally 4-connected binary M(K 3,3 )-free matroids. Mayhew and Royle [5], and independently Kingan and Lemos [7], determined all internally 4-connected bi- 2

49 50 51 52 53 54 55 nary Prism-free (therefore M(K 5 \e)-free) matroids. For each matroid N in the above list with exactly nine elements, the problem of characterizing 3- connected binary matroids with no N-minor is still unsolved yet. The problem of characterizing internally 4-connected binary AG(3, 2)-free matroids is also open. Since Z 4 has an AG(3, 2)-minor, characterizing internally 4-connected binary Z 4 -free matroids is an even harder problem. Oxley [8] determined all 3-connected binary matroids with no P 9 - or P9 -minor: 56 Theorem 1.1. Let M be a binary matroid. Then M is 3-connected having 57 no minor isomorphic to P 9 or P9 if and only if (i) M is regular and 3-connected; 58 59 60 (ii) M is a binary spike Z r, Z r, Z r \y r or Z r \t for some r 4; or (iii) M = F 7 or F 7. 61 P 9 is a very important matroid and it appears frequently in the structural 62 matroid theory (see, for example, [4, 8, 13]). In this paper, we give a complete 63 characterization of the 3-connected binary matroids with no P 9 -minor. Before 64 we state our main result, we describe a class of non-regular matroids. First 65 let K be the class 3-connected cographic matroids N = M (K 3,n ), M (K 3,n ), 66 M (K 3,n (K 3,n 3,n, K 3,n, and K 3,n 67 are obtained from K 3,n by adding one, two, or three edges in the color class of size three, respectively. Note that when n = 2, N 68 = W 4, or the cycle matroid 69 of the prism graph. From now on, we will use Prism to denote the prism 70 graph as well as its cycle matroid. Take any t disjoint triangles T 1, T 2,..., T t 71 (1 t n) of N and t copies of F 7. Perform 3-sum operations consecutively 72 starting from N and F 7 along the triangles T i (1 i t). Any resulting 73 matroid in this infinite class of matroids is called a (multi-legged) starfish. 74 Note that each starfish is not regular since at least one Fano was used (and 75 therefore the resulting matroid has an F 7 -minor) in the construction. The 76 class of starfishes and the class of spikes have empty intersection as spikes are 77 W 4 -free, while each starfish has a W 4 -minor. 78 79 80 81 Our next result, the main result of this paper, generalizes Oxley s Theorem 1.1 and completely determines the 3-connected P 9 -free binary matroids. The matroid Y 16, a single-element extension of P G(3, 2), in standard representation without the identity matrix is given in Figure 2. 82 83 84 85 Theorem 1.2. Let M be a binary matroid. Then M is 3-connected having no minor isomorphic to P 9 if and only if one of the following is true: (i) M is one of the 16 internally 4-connected non-regular minors of Y 16 ; or (ii) M is regular and 3-connected; or 3

86 87 (iii) M is a binary spike Z r, Z r, Z r \y r or Z r \t for some r 4; or (iv) M is a starfish. 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Figure 2: A binary standard representation for Y 16 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 The next result, which follows easily from the last theorem, characterizes all binary P 9 -free matroids. Theorem 1.3. Let M be a binary matroid. Then M has no minor isomorphic to P 9 if and only if M can be constructed from internally 4-connected non-regular minors of Y 16, 3-connected regular matroids, binary spikes, and starfishes using the operations of direct sum and 2-sum. Proof. Since every matroid can be constructed from 3-connected proper minors of itself by the operations of direct sum and 2-sum, by Theorem 1.2, the forward direction is true. Conversely, suppose that M = M 1 M 2, or M = M 1 2 M 2, where M 1 and M 2 are both P 9 -free. As P 9 is 3-connected, by [9, Proposition 8.3.5], M is also P 9 -free. Thus if M is constructed from internally 4-connected non-regular minors of Y 16, 3-connected regular matroids, binary spikes, and starfishes using the operations of direct sum and 2-sum, then M is also P 9 -free. Our proof does not use Theorem 1.1 except we use the fact that all spikes are P 9 -free which can be proved by an easy induction argument. In Section 2, we determine all internally 4-connected binary P 9 -free matroids. These are exactly the 16 internally 4-connected non-regular minors of Y 16. These matroids are determined using the Sage matroid package and the computation is confirmed by the matroid software Macek. Most of the work is in Section 3, which is to determine how the internally 4-connected pieces can be put together to avoid a P 9 -minor. 4

110 111 For terminology we follow [9]. Let M be a matroid. The connectivity function λ M of M is defined as follows. For X E let λ M (X) = r M (X) + r M (E X) r(m). (1) 112 113 114 115 116 117 118 119 120 Let k Z +. Then both X and E X are said to be k-separating if λ M (X) = λ M (E X) < k. If X and E X are k-separating and min{ X, E X } k, then (X, E X) is said to be a k-separation of M. Let τ(m) = min{ j : M has a j-separation} if M has a k-separation for some k; otherwise let τ(m) =. M is k-connected if τ(m) k. Let (X, E X) be a k-separation of M. This separation is said to be a minimal k-separation if min{ X, E X } = k. The matroid M is called internally 4-connected if and only if M is 3- connected and the only 3-separations of M are minimal (in other words, either X or Y is a triangle or a triad). 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 2 Characterizing internally 4-connected binary P 9 - free matroids In this section, we determine all internally 4-connected binary P 9 -free matroids. Theorem 2.1. A binary matroid M is internally 4-connected and P 9 -free if and only if (i) M is internally 4-connected graphic or cographic; or (ii) M is one of the 16 internally 4-connected non-regular minors of Y 16 ; or (iii) M is isomorphic to R 10. Sandra Kingan recently informed us that she also obtained the internally 4-connected binary P 9 -free matroids as a consequence of a decomposition result for 3-connected binary P 9 -free matroids. The following two well-known theorems of Seymour [11] will be used in our proof. Theorem 2.2. (Seymour s Splitter Theorem) Let N be a 3-connected proper minor of a 3-connected matroid M such that E(N) 4 and if N is a wheel, it is the largest wheel minor of M; while if N is a whirl, it is the largest whirl minor of M. Then M has a 3-connected minor M which is isomorphic to a single-element extension or coextension of N. Theorem 2.3. If M is an internally 4-connected regular matroid, then M is graphic, cographic, or is isomorphic to R 10. 5

143 144 145 146 The following result is due to Zhou [13, Corollary 1.2]. Theorem 2.4. A non-regular internally 4-connected binary matroid other than F 7 and F7 contains one of the following matroids as a minor: N 10, K 5, K 5, T12 \e, and T 12 /e. 148 147 The matrix representations of these matroids can be found in [13]. We use X 10 to denote the matroid K 5. It is straightforward to verify that among the 149 five matroids in Theorem 2.4, only X 10 has no P 9 -minor. We use L to denote 150 the set of matroids consisting of the following matroids in reduced standard 151 representation, in addition to F 7, F7 and Y 16. From the matrix representations 152 of these matroids, it is straightforward to check that each matroid in L is a 153 minor of Y 16, and each has an X 10 -minor. Indeed, It is clear that (i) each 154 X i is a single-element co-extension of X i 1 for 11 i 15; (ii) each Y i is 155 a single-element extension of X i 1 for 11 i 16; (iii) each Y i is a single- 156 element co-extension of Y i 1 for 11 i 16, and it is easy to check that (iv) 157 in the list X 10, X 11, X 12, X 13, each matroid is a single-element coextension of 158 its immediate predecessor. Therefore, X 10 is a minor of all matroids in L, and 159 each is a minor of Y 16. From these matrices, it is also routine to check that 160 the only matroid of L having a triangle is F 7 (this can also be easily verified 161 by using the Sage matroid package). X 10 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 X 11 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 X 11 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 Y 11 : 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 X 12 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 X 12 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 1 1 1 Y 12 : 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 6

X 13 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 Y 13 : 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 X 14 : 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0 Y 14 : 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 1 X 15 = P G(3, 2) : 0 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 Y 15 : 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 (2) 171 162 Proof of Theorem 2.1: If M is one of the matroids listed in (i) to (iii), then 163 M is internally 4-connected. All matroids in (i) or (iii) are regular, thus are 164 P 9 -free. Using the Sage matroid package, it is easy to verify that Y 16 is P 9 -free, 165 hence all matroids in (ii) are also P 9 -free. Let M be an internally 4-connected 166 binary matroid with no P 9 -minor. If M is regular, then by Theorem 2.3, M is 167 either graphic, cographic, or isomorphic to R 10, which is regular. Therefore, 168 we need only show that an internally 4-connected matroid M is non-regular 169 and P 9 -free if and only if M is a non-regular minor of Y 16. Suppose that M is 170 an internally 4-connected non-regular and P 9 -free matroid. If M has exactly seven elements, then M = F 7 or M = F7. Suppose that M has at least eight 172 elements. By Theorem 2.4, M has an N 10, X 10, X10 12\e, or T 12 /e-minor. 173 Since all but X 10 has a P 9 -minor among these matroids, M must have an X 10-174 minor. We use the Sage matroid package (by writing simple Python scripts) 175 and the matroid software Macek independently to do our computation and 176 have obtained the same result. Excluding P 9, we extend and coextend X 10 177 seven times and found only thirteen 3-connected binary matroids. These matroids are X 11, X 11, Y 11, X 12, X 12, Y 12, X 13, Y 13, X 14, Y 14, X 15 178 = P G(3, 2), Y 15, 7

179 180 181 182 183 184 and Y 16 ; each having at most 16 elements; each being a minor of Y 16 ; and each being internally 4-connected. As X 10 is neither a wheel nor a whirl, by the Splitter Theorem (Theorem 2.2), M is one of the matroids in L, each of which is a non-regular internally 4-connected minor of Y 16. Note that all non-regular internally 4-connected minors of Y 16 are P 9 -free, hence L consists of all internally 4-connected non-regular minors of Y 16. 185 3 Characterizing 3-connected binary P 9 -free matroids 186 In this section, we will prove our main result. We begin with several lemmas. 187 Let G be a graph with a specified triangle T = {e 1, e 2, e 3 }. By a rooted K 4 - minor using T we mean a loopless minor H of G such that si(h) 188 = K 4 ; 189 {e 1, e 2, e 3 } remains a triangle of H; and H\{e i, e j } is isomorphic to K 4, for 190 some distinct i, j {1, 2, 3}. By a rooted K 4 -minor using T we mean a loopless minor H of G such that si(h) 191 = K 4 ; {e 1, e 2, e 3 } remains a triangle of H; and 192 H\e i is isomorphic to K 4, for some i {1, 2, 3}. Let T be a specified triangle 193 of a matroid M. We can define a rooted M(K 4 )-minor using T and a rooted 194 M(K 4 )-minor using T similarly. Moreover, in the following proof, any K 4 195 is obtained from K 4 by adding a parallel edge to an element in the common 196 triangle T used in the 3-sum specified in the context. 197 198 199 200 201 202 203 Lemma 3.1. ([12]) Let T be a triangle of 3-connected binary matroid M with at least four elements. Then T is contained in a M(K 4 )-minor of M. Lemma 3.2. ([1]) Let T be a triangle of a binary non-graphic matroid M. Then the following are true: (i) If M is non-regular, then T is contained in a F 7 -minor; (ii) If M is regular but not graphic, then T is contained in a M (K 3,3 )- minor. 204 205 206 207 208 209 210 211 212 Let M 1 and M 2 be matroids with ground sets E 1 and E 2 such that E 1 E 2 = T and M 1 T = M 2 T = N. The following result of Brylawski [2] about the generalized parallel connection can be found in [9, Propsition 11.4.14]. Lemma 3.3. The generalized parallel connection P N (M 1, M 2 ) has the following properties: (i) P N (M 1, M 2 ) E 1 = M 1 and P N (M 1, M 2 ) E 2 = M 2. (ii) If e E 1 T, then P N (M 1, M 2 )\e = P N (M 1 \e, M 2 ). (iii) If e E 1 cl 1 (T ), then P N (M 1, M 2 )/e = P N (M 1 /e, M 2 ). (iv) If e E 2 T, then P N (M 1, M 2 )\e = P N (M 1, M 2 \e). 8

213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 (v) If e E 2 cl 2 (T ), then P N (M 1, M 2 )/e = P N (M 1, M 2 /e). (vi) If e T, then P N (M 1, M 2 )/e = P N/e (M 1 /e, M 2 /e). (vii) P N (M 1, M 2 )/T = (M 1 /T ) (M 2 /T ). In the rest of this paper, we consider the case when the generalized parallel connection is defined across a triangle T, where T is the common triangle of the binary matroids M 1 and M 2. Then P N (M 1, M 2 ) = P N (M 2, M 1 ) (see [9, Propsition 11.4.14]). Moreover, N = M 1 T = M 2 T = U 2,3. We will use T to denote both the triangle and the submatroid M 1 T. Thus we use P T (M 1, M 2 ) instead of P N (M 1, M 2 ) for the rest of the paper. Lemma 3.4. Let M = P T (M 1, P S (M 2, M 3 )) where M i is a binary matroid (1 i 3); S is the common triangle of M 2 and M 3 ; T is the common triangle of M 1 and M 2. Then the following are true: (i) if E(M 1 ) (E(M 3 )\E(M 2 )) =, then M = P S (P T (M 1, M 2 ), M 3 ); (ii) if E(M 1 ) E(M 3 ) =, then M 1 3 (M 2 3 M 3 ) = (M 1 3 M 2 ) 3 M 3. Proof. (i) As E(M 1 ) (E(M 3 )\E(M 2 )) =, T = E(M 1 ) E(P S (M 2, M 3 )), and T is the common triangle of M 1 and P S (M 2, M 3 ). Moreover, S = E(M 3 ) E(P T (M 1, M 2 )), and S is the common triangle of M 3 and P T (M 1, M 2 ). By [9, Proposition 11.4.13], a set F of M is a flat if and only if F E(M 1 ) is a flat of M 1 and F E(P S (M 2, M 3 )) is a flat of P S (M 2, M 3 ). The latter is true if and only if [F (E(M 2 ) E(M 3 ))] E(M i ) = F E(M i ) is a flat of M i for i = 2, 3. Therefore, F is a flat of M if and only if F E(M i ) is a flat of M i for 1 i 3. The same holds for P S (P T (M 1, M 2 ), M 3 ). Thus M = P S (P T (M 1, M 2 ), M 3 ). (ii) As E(M 1 ) E(M 3 ) =, we deduce that S T =, and the conclusion of (i) holds. Therefore, 235 236 237 238 239 P T (M 1, P S (M 2, M 3 ))\(S T ) = P S (P T (M 1, M 2 ), M 3 )\(S T ). By Lemma 3.3, we conclude that P T (M 1, P S (M 2, M 3 )\S)\T = P S (P T (M 1, M 2 )\T, M 3 )\S. That is, M 1 3 (M 2 3 M 3 ) = (M 1 3 M 2 ) 3 M 3. Lemma 3.5. Let M = P T (M 1, M 2 ) where M i is a binary matroid (1 i 2) and T is the common triangle of M 1 and M 2. Then C is a cocircuit of M if and only if one of the following is true: (i) C is a cocircuit of M 1 or M 2 avoiding T ; 240 (ii) C = C1 C 2 where C i is a cocircuit of M i such that C1 T = C 2 T, 241 which has exactly two elements. 9

Proof. By [9, Proposition 11.4.13], a set F of M is a flat if and only if F E(M i ) is a flat of M i for 1 i 2. Moreover, for any flat F of M, r(f ) = r(f E(M 1 ))+r(f E(M 2 )) r(f T ) (see, for example, [9, (11.23)]). Let C be a cocircuit of M and H = E(M) C. As M is binary, C T = 0, 2, and thus H T = 3, 1. First assume that C T = 0. As r(h) = r(h E(M 1 ))+r(h E(M 2 )) r(h T ), then r(m) 1 = r(m 1 )+r(m 2 ) 3 = r(h) = r(h E(M 1 )) + r(h E(M 2 )) 2. Thus, r(m 1 ) + r(m 2 ) 1 = r(h E(M 1 )) + r(h E(M 2 )). 242 243 244 245 246 Therefore, either r(h E(M 1 )) = r(m 1 ) 1 and r(h E(M 2 )) = r(m 2 ), or r(h E(M 2 )) = r(m 2 ) 1 and r(h E(M 1 )) = r(m 1 ). In the former case, as H E(M 1 ) and H E(M 2 ) are flats of M 1 and M 2 respectively, we deduce that H E(M 2 ) = E(M 2 ); H E(M 1 ) is a hyperplane of M 1 and thus C E(M 1 ) is a cocircuit of M 1 avoiding T. The latter case is similar. If C T = 2, then H T = 1. As r(h) = r(h E(M 1 )) + r(h E(M 2 )) r(h T ), we deduce that r(m) 1 = r(m 1 ) + r(m 2 ) 3 = r(h) = r(h E(M 1 )) + r(h E(M 2 )) 1. We conclude that r(m 1 ) + r(m 2 ) 2 = r(h E(M 1 )) + r(h E(M 2 )). 247 Now, for 1 i 2, H E(M i ) is a proper flat of M i, so that r(h 248 E(M i )) r(m i ) 1. Therefore, r(h E(M 1 )) = r(m 1 ) 1 and r(h 249 E(M 2 )) = r(m 2 ) 1. We conclude that Ci = E(M i) H is a cocircuit of M i 250 and C = C1 C 2 such that C 1 T = C 2 T, which has exactly two elements. 251 Note that the converse of the above arguments is also true, thus the proof of 252 the lemma is complete. 253 254 255 256 The following corollary might be of independent interest. Corollary 3.6. Let M 1 and M 2 be a binary matroids and M = M 1 3 M 2 such that M 1 and M 2 have the common triangle T. Then the following are true: 257 (i) any cocircuit C of M is either a cocircuit of M 1 or M 2 avoiding T, or 258 C = C1 C 2 where C i is a cocircuit of M i (i = 1, 2) such that C1 T = C 2 T, 259 which has exactly two elements. 262 260 (ii) if C is either a cocircuit of M 1 or M 2 avoiding T, then C is also 261 a cocircuit of M. Moreover, suppose that Ci is a cocircuit of M i such that C1 T = C 2 T, which has exactly two elements. Then either C 1 C 2 is a 263 cocircuit of M, or C1 C 2 is a disjoint union of two cocircuits R and Q of 264 M, where R and Q meet both M 1 and M 2. 10

271 265 Proof. As M = M 1 3 M 2 = P T (M 1, M 2 )\T, the cocircuits of M are the 266 minimal non-empty members of the set F = {D T : D is a cocircuit of 267 P T (M 1, M 2 )}. If C is a cocircuit of M, then C = D T for some cocircuit 268 D of P T (M 1, M 2 ). By the last lemma, either (a) D is a cocircuit of M 1 or M 2 269 avoiding T, or (b) D = C1 C 2 where C i is a cocircuit of M i (i = 1, 2) such 270 that C1 T = C 2 T, which has exactly two elements. In (a), C = D, and in (b), C = C1 C 2. Hence either (i) or (ii) holds in the lemma. 272 Conversely, if C is either a cocircuit of M 1 or M 2 avoiding T, then clearly 273 C is also a cocircuit of M, as C = C T is clearly a non-empty minimal 274 member of the set F. Now suppose that Ci (i = 1, 2) is a cocircuit of M i such that C1 T = C 2 T, which has exactly two elements. If C 1 C 2 is not a 276 cocircuit of M, then it contains a cocircuit R of M which is a proper subset 277 of C1 C 2. Clearly, R must meet both C1 and C 2. By (i), R = R1 R 2, 278 where Ri is a cocircuit of M i (i = 1, 2) such that R1 T = R 2 T, which 1 2 T = {x, y}, then 280 R1 2 T = {x, z} or {y, z}, say the former. Moreover, R i \T is a 281 proper subset of Ci \T for i = 1, 2 as T does not contain any cocircuit of 282 either M 1 or M 2. As both M 1 and M 2 are binary, Q i = Ci R i (i = 1, 2) 283 contains, and indeed, is a cocircuit of M i such that Q 1 T = Q 2 T = {y, z}. 284 Now it is straightforward to see that Q 1 Q 2 is a minimal non-empty member 285 of F and thus is a cocircuit of M. As C = R Q, (ii) holds. 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 The 3-sum of two cographic matroids may not be cographic. However, the following is true. Lemma 3.7. Suppose that M 1 = M (G 1 ) and M 2 = M (G 2 ) are both cographic matroids with u and v being vertices of degree three in G 1 and G 2, respectively. Label both uu i and vv i as e i (1 i 3) so that T = E(M 1 ) E(M 2 ) = {e 1, e 2, e 3 } is the common triangle of M 1 and M 2. Then P T (M 1, M 2 ) = M (G), where G is obtained by adding a matching {u 1 v 1, u 2 v 2, u 3 v 3 } between G 1 u and G 2 v. In particular, M (G 1 ) 3 M (G 2 ) = M (G/e, f, g) is also cographic. Proof. We need only show that P T (M 1, M 2 ) and M (G) have the same set of cocircuits. By Lemma 3.5, C is a cocircuit of M = P T (M 1, M 2 ) if and only if one of the following is true: (i) C is a cocircuit of M 1 or M 2 avoiding T. In other words, C is either a circuit of G 1 or a circuit of G 2 which does not meet T (i.e., C is a circuit of either G 1 u or a circuit of G 2 v); 301 (ii) C = C1 C 2 where C i is a cocircuit of M i such that C1 T = 302 C2 T, which has exactly two elements. In other words, C = C1 C 2 where 303 C (i = 1, 2) is a circuit of G i containing u and v respectively, such that i 11

304 C1 T = C 2 T, which contains exactly two edges. Now it is easily seen 305 that the set of cocircuits of M is exactly equal to the set of circuits of M(G) 306 (or the set of cocircuits of M (G)). In particular, M (G 1 ) 3 M (G 2 ) = 307 P T (M (G 1 ), M (G 2 ))\T = M (G)\T = M (G/e, f, g) is cographic. This 308 completes the proof of the lemma. 309 310 311 312 313 The following consequence of the last lemma will be used frequently in the paper. Corollary 3.8. Suppose that M (K 3,m ), M (K 3,m ), M (K 3,n ) K (m, n 2). Then the following are true: (i) M (K 3,m ) 3 M (K 3,n )) = M (K 3,m+n 2 ); (ii) M (K 3,m ) 3 M (K 3,n ) 314 = M (K 3,m+n 2 ); 315 (iii) P (M (K 3,m), M(K 4 )) is cographic and is isomorphic to M (G) where 316 317 G is obtained by putting a 3-edge matching between the 3-partite set of K 3,m 1 and the three vertices of K 3. 318 319 320 (iv) M (K 3,m ) 3 M(K 4 )) = M (K 3,m ) where K 4 is obtained from K 4 by adding a parallel edge to an element in the common triangle T used in the 3-sum. 321 (v) if M 1 = M (K 3,m ), and M 2 = M(K 4 ), then depending on which 322 element in T is in a parallel pair in M(K 4 ) and which extra edge was added 323 to K 3,m 3,m, the matroid M 1 3 M 2 is either isomorphic to M (K 3,m 324 or M (G), where G is obtained from K 3,m by adding an edge parallel to the 325 extra edge. 326 327 328 (vi) if M 1 K and M 2 = M(K 4 ), then either M 1 3 M 2 K or M 1 3 M 2 = M (G), where G has a parallel pair which does not meet any triad of G. 329 (vii) if M 1 K and M 2 K, then either M 1 3 M 2 K or M 1 3 M 2 = 330 M (G), where G has at least one parallel pair which does not meet any triad 331 of G. 333 332 Proof. (i)-(v) are direct consequences of Lemma 3.7. Suppose that M 1 K and is isomorphic to M (K 3,m ), M (K 3,m ), M (K 3,m ), or M (K 3,m ). Then 334 either M 1 3 M 2 = M (K 3,m (K 3,m ) or M (K 3,m ) and thus is in K (in 335 this case, M 1 is not isomorphic to M (K 3,m )), or isomorphic to M (G), where G is obtained from K 3,m, K 3,m, or K 3,m by adding an edge in parallel to an 337 existing edge added between two vertices of the 3-partite set of K 3,m. Clearly, 338 this parallel pair does not meet any triad of G. We omit the straightforward 339 and similar proof of (vii). 12

340 Corollary 3.9. Let M be a binary matroid and M = M 1 3 M 2 where M 1 is a starfish. Suppose that M 2 is a starfish, or M 2 = M(K 4 ), or M 2 341 = M (G) K: G 342 = K 3,n, K 3,n, K 3,n, or K 3,n (n 2). Then either M is also a starfish, or 343 M has a 2-element cocircuit which does not meet any triangle of M. 344 Proof. Suppose that the starfish M 1 uses s Fano matroids and M 2 uses t 345 Fano matroids where s 1 and t 0. Clearly, in the starfish M 1, any triangle is a triad in the corresponding 3-connected graph G 1 = K3,m, K 3,m, 347 K 3,m, or K 3,m (m 2) used to construct M 1. We assume that first s = 1 and t = 0. Then by the definition of the starfish, M 1 348 = F7 3 N 1, where N 1 = M (G 1 ), and either M 2 = M(K 4 ), or M 2 349 = M (G); G is 3-connected where G = K 3,n, K 3,n, K 3,n, or K 3,n (n 2). By Lemma 3.4, we have that 351 2 = F7 3 (N 1 3 M 2 ) (the condition of the lemma is 352 clearly satisfied). By Corollary 3.8, we deduce that either N 1 3 M 2 K, or 353 it has a 2-element cocircuit avoiding any triangle of N 1 3 M 2. In the former 354 case, we conclude that M is a starfish. In the latter case, by Corollary 3.6, M 355 has a 2-element cocircuit avoiding any triangle of M. The general case follows 356 from an easy induction argument using Lemmas 3.4 and Corollaries 3.6 and 357 3.8. Lemma 3.10. Suppose that M = M (G) for a 3-connected graph G 358 = K 3,n, 359 K 3,n, K 3,n, or K 3,n (n 2), or M is a starfish. Then for any triangle T of 360 M, there are at least two elements e 1, e 2 of T, such that for each e i (i = 1, 2), 361 there is a rooted K 4 -minor using both T and e i such that e i is in a parallel 362 pair. Proof. Suppose that M = M (G) for a 3-connected graph G 363 = K 3,n, K 3,n, 364 K 3,n, or K 3,n (n 2). When n 3, the proof is straightforward. When n = 2, then G 365 = W 4 or K 5 \e, and the result is also true. Now suppose that M is a starfish constructed by starting from N 366 = M (G) for a 3-connected graph G 367 = K 3,n, K 3,n, K 3,n, or K 3,n (n 2) with t (1 368 t n) copies of F 7 by performing 3-sum operations. Choose an element f i 369 of E(M) in each copy of F 7 (1 i t). By the definition of a starfish, and 370 by using Lemma 3.3(iii),(v), M/f 1, f 2,... f t is isomorphic to N containing T. 371 Now the result follows from the above paragraph. 372 373 374 375 376 We will need the following result [11, 11.1]. Lemma 3.11. Let e be an edge of a simple 3-connected graph G on more than four vertices. Then either G\e is obtained from a simple 3-connected graph by subdividing edges or G/e is obtained from a simple 3-connected graph by adding parallel edges. 13

377 378 379 380 381 382 383 384 385 386 387 Let G = (V, E) be a graph and let x, y be distinct elements of V E. By adding an edge between x, y we obtain a graph G defined as follows. If x and y are both in V, we assume xy E and we define G = (V, E {xy}); if x is in V and y = y 1 y 2 is in E, we assume x {y 1, y 2 } and we define G = (V {z}, (E\{y}) {xz, y 1 z, y 2 z}); if x = x 1 x 2 and y = y 1 y 2 are both in E, we define G = (V {u, v}, (E\{x, y}) {ux 1, ux 2, uv, vy 1, vy 2 }) Lemma 3.12. Let G be a simple 3-connected graph with a specified triangle T. Then G has a rooted K 4 -minor unless G is K 4, W 4, or Prism. Proof. Suppose the Lemma is false. We choose a counterexample G = (V, E) with E as small as possible. Let x, y, z be the vertices of T. We first prove that G {x, y, z} has at least one edge. 388 Suppose G {x, y, z} is edgeless. Since G is 3-connected, every vertex 389 in V {x, y, z} must be adjacent to all three of x, y, z, which means that 390 G = K 3,n for a positive integer n. Since G is a counterexample, G cannot 391 be K 4 and thus G contains K 3,2, which contains a rooted K 4 -minor. This 392 contradicts the choice of G and thus G {x, y, z} has at least one edge. 393 394 395 396 Let e = uv be an edge of G {x, y, z}. By Lemma 3.12, there exists a simple 3-connected graph H such that at least one of the following holds: Case 1. G\e is obtained from H by subdividing edges; Case 2. G/e is obtained from H by adding parallel edges. 397 Since H is a proper minor of G and H still contains T, by the minimality of 398 G, H has to be K 4, W 4, or Prism, because otherwise H (and G as well) would 399 have a rooted K 4 -minor. Now we need to deduce a contradiction in Case 1 400 and Case 2 for each H {K 4, W 4, Prism}. 403 401 Let P + be obtained from Prism by adding an edge between two nonadja- 402 cent vertices. Before we start checking the cases we make a simple observation: with respect to any of its triangles, P + has a rooted K 4 -minor. As a result, G 404 cannot contain a rooted P + -minor: a P + -minor in which T remains a triangle. 405 406 407 408 409 410 411 412 413 414 We first consider Case 1. Note that G is obtained from H by adding an edge between some α, β V E. By the choice of e, we must have α, β V (T ) E(T ). If H = K 4 then G = P rism, which contradicts the choice of G. If G = W 4 or P rism, then it is straightforward to verify that G contains a rooted P + -minor (by contracting at most two edges), which is a contradiction by the above observation. Next, we consider Case 2. Let w be the new vertex created by contracting e. Then G/e is obtained from H by adding parallel edges incident with w. Observe that w has degree three in H, for each choice of H. Consequently, as G is simple, G has four, three, or two more edges than H. Suppose G has 14

415 four or three more edges than H. Then H is G u or G v. Without loss of 416 generality, let H = G u. Choose three paths P x, P y, P z in H from v to x, y, z, 417 respectively, such that they are disjoint except for v. Now it is not difficult 418 to see that a rooted K 4 -minor of G can be produced from the union of the 419 triangle T, the three paths P x, P y, P z, and the star formed by edges incident 420 with u. This contradiction implies that G has exactly two more edges than 421 H. Equivalently, G is obtained from H by adding an edge between a neighbor 422 s of w and an edge wt with t s. 423 If H = K 4 then G = W 4, which contradicts the choice of G. If H = W 4 424 then G = W 5 or P +. In both cases, G contain a rooted K 4 -minor, no matter 425 where the special triangle is. Finally, if H = P rism then G contains a rooted 426 P + -minor, which is impossible by our early observation. In conclusion, Case 427 2 does not occur, which completes our proof. 428 429 430 431 Lemma 3.13. Let M = M (G) be a 3-connected cographic matroid with a specified triangle T. Then M has a rooted K 4 -minor using T unless G = K 3,n, K 3,n, K 3,n, or K 3,n for some n 1. In particular, if M (G) is not graphic, then n 3. 432 Proof. Suppose that M does not contain rooted K 4 -minor using T. Note that 433 M (G) does not have a rooted K 4 -minor using T if and only if G does not have 434 a minor obtained from K 4 (where T is cocircuit) by subdividing two edges of 435 T. Now we show that T is a vertex triad (which corresponds to a star of degree 436 three). Otherwise, let G E(T ) = X Y, where T is a 3-element edge-cut but not a vertex triad. If G 437 = P rism, then clearly M (G) has a rooted K 4 -minor; 438 a contradiction. If G is not isomorphic to a Prism, we can choose a cycle in 439 one side and a vertex in another side which is not incident with any edge of T. 440 Then we can get a rooted K 4 -minor; a contradiction again. Hence the edges of 441 T are all incident to a common vertex v of degree three with neighbors v 1, v 2, 442 and v 3. A rooted K 4 -minor using T exists if and only if G has a cycle missing 443 v and at least two of v 1, v 2, and v 3. Hence every cycle of G v contains at 444 least two of v 1, v 2, and v 3, and thus G v v i v j is a tree for 1 i j 3. 445 Moreover, G v v 1 v 2 v 3 has to be an independent set. Otherwise, it is a 446 forest. Take two pedants in a tree, each of which has at least two neighbors in 447 v 1, v 2, or v 3. Thus G v contains a cycle missing at least two vertices of v 1, v 2, 448 and v 3. This contradiction shows that G v v 1 v 2 v 3 is an independent 449 set and thus G is K 3,n, K 3,n, K 3,n, or K 3,n for some n 1. In particular, if M (G) is not graphic, then n 3. 450 451 452 453 Lemma 3.14. Let M be a 3-connected binary P 9 -free matroid and M = M 1 3 M 2 where M 1 is non-regular, and M 1 and M 2 have the common triangle T. Then 15

454 455 456 457 458 (i) if M 2 is graphic, then either M 2 = M(G) where G is W4 or the Prism, or M 2 = M(K 4 ) where M(K 4 ) is obtained from M(K 4) (which contains T ) by adding an element parallel to an element of T ; (ii) if M 2 is cographic but not graphic, then M 2 = M (G), where G = K 3,n, K 3,n, K 3,n, or K 3,n for some n 3. 459 460 461 462 463 464 465 Proof. Suppose that M = P (M 1, M 2 )\T, where T is the common triangle of M 1 and M 2. As M is 3-connected, by [11, 4.3], both si(m 1 ) and si(m 2 ) are 3-connected, and only elements of T can have parallel elements in M 1 or M 2. Then by Lemma 3.2, T is contained in a F 7 -minor in si(m 1 ). Now M 2 does not contain a rooted K 4 -minor using T, where K 4 is obtained from this K 4 by adding a parallel element to any two of the three elements of T (otherwise, the 3-sum of M 1 and M 2 contains a P 9 -minor). If M 2 is graphic, then by Lemma 3.12, si(m 2 ) 466 = M(G) where G is either 467 W 3, W 4 or the Prism. When G is either W 4 or the Prism, then it is easily seen that M 2 has to be simple, and thus M 2 = W4 or Prism. If G 468 = W 3, then as M 469 is P 9 -free and M 2 has at least seven elements (from the definition of 3-sum), it is easily seen that M 2 470 = M(K 4 ). If M 2 is cographic but not graphic, then by Lemma 3.13, si(m 2 ) 471 = M (G), 472 where G is K 3,n, K 3,n, K 3,n, or K 3,n for some n 3. If M 2 is not simple, then it is straightforward to find a rooted M(K 473 4 )-minor using T in M 2, thus 474 a P 9 -minor in M; a contradiction. This completes the proof of the lemma. 475 Lemma 3.15. Let M be a 3-connected regular matroid with at least six ele- 476 ments and T be a triangle of M. Then M has no rooted M(K 4 )-minor using T if and only if M is isomorphic to a 3-connected matroid M (K 3,n ), M (K 477 478 M (K 3,n ), M (K 3,n ) for some n 1. 3,n ), 479 Proof. If M is isomorphic to a 3-connected matroid M (K 3,n ), M (K 3,n ), 480 M (K 3,n ), M (K 3,n ) (n 1), then it is straightforward to check for any triangle T, M has no rooted M(K )-minor using T. 481 4 482 Conversely, suppose that M is a 3-connected regular matroid with at least 483 six elements and T is a triangle of M, such that M has no rooted M(K 4 )- 484 minor using T. If M is internally 4-connected, then by Theorem 2.3, M is 485 486 487 488 489 490 491 either graphic, cographic, or is isomorphic to R 10. The result follows from Lemmas 3.12 and 3.13, and the fact that R 10 is triangle-free. So we may assume that M is not internally 4-connected and has a 3-separation (X, Y ) where X, Y 4. We may assume that X T 2. Suppose that Y T has exactly one element e. Then as T is a triangle, (X e, Y \e) is also a 3-separation. If Y = 4, then Y e is a triangle or a triad. Moreover, r(y ) + r (Y ) Y = 2. As M is 3-connected and binary, 16

503 492 r(y ), r (Y ) 3, and thus r(y ) = r (Y ) = 3. If Y e is a triangle, then 493 it is not a triad, and thus Y contains a cocircuit which contains e. This is 494 a contradiction as this cocircuit meets T with exactly one element. Hence 495 Y e is a triad, and from r(y ) = 3, there is an element f T, f e 496 such that Y f is a triangle. In other words, Y forms a 4-element fan. We conclude that M = M 1 3 M(K 4 ) by [11, 2.9] where S is the common triangle 498 of M 1 and M(K 4 4 ) is obtained from M(K 4) (containing T ) by 499 adding an element e 1 in parallel to an element e of S. By switching the 500 label of e 1 to e in M 1, we obtain a matroid M 1 ( = M 1 ) which is isomorphic 501 to a minor of M having triangle T. By [11, 4.3], si(m 1 ) is 3-connected. 502 Hence by induction, si(m 1 ) is isomorphic to a 3-connected matroid M (K 3,m ), M (K 3,m ), M (K 3,m (K 3,m ) for some m 1. As M has no rooted 504 M(K 4 )-minor M 1 (S T ) > 2. Moreover, the element 505 e 1 is in two triangles of si(m 1 ), so m 3. Now using Lemma 3.7, it is straightforward to verify that M = W 4 = M (K 3,2 ) and thus the Lemma 507 holds. Hence we may assume that Y 5 and thus Y \e 4. 508 Therefore we may assume that M has a separation (X, Y ) such that T 509 X, and both X and Y have at least four elements. Hence by [11, (2.9)], 510 M = M 1 3 M 2 where M 1 and M 2 are isomorphic to minors of M having the 511 common triangle S, and T is a triangle of M 1. Moreover, E(M i ) < E(M) 512 for i = 1, 2, and both si(m 1 ) and si(m 2 ) are 3-connected [11, (4.3)]. First 513 assume that each element of S is parallel to an element of T in M 1. Then by 514 Lemma 3.1, si(m 1 ) contains a rooted M(K 4 )-minor using T. As each element 515 of T in M 1 is in a parallel pair, we conclude that M has a rooted M(K 4 )-minor 516 using T ; a contradiction. 517 So we may assume that at least one element of T is not parallel to an 518 element of S (as M is binary, there are at least two such elements). As 519 si(m 1 ) is a 3-connected minor of M, it has no rooted M(K 4 )-minor using T. By induction, si(m 1 ) = M (K 3,s ), M (K 3,s (K 3,s (K 3,s ) for some 521 = M(K 4 ). Remove all elements of M 1 not in the set S T 522 in P S (M 1, M 2 ). Then every element of T \S is parallel to an element of S\T. 523 Contracting all elements of S\T, we obtained a minor of M isomorphic to M 2 and T is a triangle of this minor. By induction again, si(m 2 ) 524 = M (K 3,t ), M (K 3,t ), M (K 3,t (K 3,t 2) 525 = M(K 4 ). Suppose that si(m i ) 526 = M(K 4 ) for some i = 1, 2. Then as M i have at least seven 527 elements and M has no rooted M(K 4 )-minor i = 528 M(K 4 ). As M has no M(K 4 )-minor containing T, and M is 3-connected, using Corollary 3.8, it is routine to verify that M = M (K 3,n ), M (K 529 530 M (K 3,n ), or M (K 3,n ) for some n 2. Corollary 3.16. Let M be a 3-connected binary non-regular P 9 -free matroid. 3,n ), 531 532 Suppose that M = M 1 3 M 2 such that M 1 and M 2 have the common triangle 17

533 T. If M 2 is regular, then M 2 is isomorphic to a 3-connected matroid M (K 3,n ), M (K 3,n ), M (K 3,n ), or M (K 3,n ) (n 2), or M 2 534 = M(K 4 ) where M(K 4 ) 535 is obtained from M(K 4 ) (containing T ) by adding an element in parallel to an 536 element of T. 541 537 Proof. As M is 3-connected, by [11, 4.3], both si(m 1 ) and si(m 2 ) are 3-538 connected, and only elements of T can have parallel elements in M 1 or M 2. 539 As M is non-regular and M 2 is regular, si(m 1 ) is non-regular and thus (by 540 Lemma 3.2) has a F 7 -minor containing the common triangle T of M 1 and M 2. As M is P 9 -free, M 2 has no rooted M(K 4 )-minor using T. By Lemma 3.15, si(m 2 ) is isomorphic to a 3-connected matroid M (K 3,n ), M (K 542 543 M (K 3,n ), M (K 3,n ) (n 2), or M(K 4). Now using Lemma 3.10, it is straightforward to check that either M 2 = M(K 544 4 ), or M 2 is simple, and M 2 545 = M (K 3,n ), M (K 3,n ), M (K 3,n ), or M (K 3,n ) (n 2). 546 Now we are ready to prove our main theorem. 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 3,n ), Proof of Theorem 1.2. Suppose that a starfish M is constructed from a 3- connected cographic matroid N by consecutively applying the 3-sum operations with t copies of F 7, where N = M (G); G = K 3,n, K 3,n, K 3,n, or K 3,n for some n 2. First we show that M is 3-connected. We use induction on t. When t = 0, N is 3-connected. Suppose that M is 3-connected for t < k n. Now suppose that t = k. Then M = M 1 3 F, where F = F 7 and M 1 and F share the common triangle T. Take an element f of E(F ) E(M). Then by Lemma 3.3, M/f = P (M 1, F/e)\T = M 1, which is a starfish with t = k 1, and thus is 3-connected by induction. If M is not 3-connected, then f is either in a loop of M, or is in a cocircut of size one or two. Clearly, M does not have any loop, thus f is in a cocircuit C of M with size one or two. As P (M 1, F ) is 3-connected, it does not contain any cocircuit of size less than three. Hence C T contains a cocircuit D of P (M 1, F ). As P (M 1, F ) is binary, D T has exactly two elements, and thus D has at most four elements. As T contains no cocircuit of either M 1 or F, by Lemma 3.5, F = F 7 has a cocircuit of size at most three meeting two elements of T. This contradiction shows that M is 3-connected. 568 564 Next we show that if M is one of the matroid listed in (i)-(iv), then M 565 is P 9 -free. By Theorem 2.1 and the fact that all spikes and regular matroids 566 are P 9 -free, we need only show that any starfish is P 9 -free. We use induction 567 on the number of elements of the starfish M. By the definition, the unique smallest starfish has nine elements, and is isomorphic to P9. Clearly, P 9 is 569 P 9 -free. Suppose that any starfish with less than n ( 10) elements is P 9 -free. 570 Now suppose that we have a starfish M with n elements. Suppose, on the 571 contrary, that M has a P 9 -minor. Then by the Splitter Theorem (Theorem 572 2.2), there is an element e in M such that either M\e or M/e is 3-connected 18

573 having a P 9 -minor. Note that the elements of a starfish consists of two types: 574 those are subsets of E(N) (denote this set by K), or those are in part of copies 575 of F 7 (denote this set by F ). Then E(M) = K F. First we assume that 576 e F. Then M = M 1 3 M 2, where M 1 is either one of M (K 3,n ), M (K 3,n ), M (K 3,n (K 3,n 2 577 = F 7, and e E(M 2 ). By the construction of the starfish and Lemma 3.3, M/e 578 = M 1 579 and is either cographic or a smaller starfish and therefore does not contain 580 a P 9 -minor; a contradiction. Therefore M\e is 3-connected and contains a P 9 -minor. But then by Lemma 3.4, M\e 581 = P (M 1, M(K 4 ))\T. By Corollary 582 3.8, as M\e is 3-connected, we conclude that M\e is a smaller starfish and 583 therefore is P 9 -free. This contradiction shows that e K. 584 If e is in a triangle of M, then M/e is not 3-connected, and thus M\e 585 is 3-connected and contains a P 9 -minor. Each triangle of M is correspond- 586 ing to a triad in G. By Lemmas 3.3 and 3.4 again, we can do the deletion 587 N\e first, then perform the 3-sum operations with copies of F 7. Note that N\e = M (G/e) where G 588 = K 3,n, K 3,n, K 3,n, or K 3,n (n 2). As M\e is 3-connected and thus simple, we deduce that n 3, N 589 = M (K 3,n ) or M (K 3,n ), and N\e 590 = M (K 3,n 1 ), or M (K 3,n 1 ). Therefore, M\e is an- 591 other starfish and does not contain any P 9 -minor by induction; a contradiction. 592 Finally assume that e K is not in any triangle of M. Then e is not in any triad of G. Hence if n = 2, then G 593 = K 3,2. As G/e has parallel elements, 594 the matroid N\e has serial-pairs, and thus M\e is not 3-connected, we con- clude that M/e is 3-connected having a P 9 -minor. Note that N 595 = M (K 3,n ), M (K 3,n ), or M (K 3,n ) (n 2), and thus N/e 596 = M (K 3,n ), M (K 3,n ), or 597 M (K 3,n ), which is still 3-connected. We conclude again, by Lemma 3.3, that 598 M/e is a smaller starfish than M, thus cannot contain any P 9 -minor. This 599 contradiction completes the proof of the first part. 600 Now suppose that M is a 3-connected binary matroid with no P 9 -minor. 601 We may assume that M is not regular. If M is internally 4-connected, then 602 the theorem follows from Theorem 2.1. Now suppose that M is neither regular 603 nor internally 4-connected. We show that M is either a spike or a starfish. 604 Suppose that E(M) 9. As M is not internally 4-connected, M is not F 7 605 or F7. Hence E(M) 8. Then M is AG(3, 2), S 8, Z 4, Z4 (all spikes), or P 9, which is the 3-sum of F 7 and W 4 = M (K 3,2 ), thus is a starfish. We conclude 607 that the result holds for E(M) 9. Now suppose that E(M) 10. As 608 M is not internally 4-connected, M = M 1 3 M 2 = P (M 1, M 2 )\T, where M 1 609 and M 2 are isomorphic to minors of M ([11, 4.1]) and T = {x, y, z} is the 610 common triangle of M 1 and M 2. Moreover, E(M i ) < E(M) for i = 1, 2, 611 and both si(m 1 ) and si(m 2 ) are 3-connected [11, (4.3)]. The only possible 612 parallel element(s) of either M 1 or M 2 are those in the common triangle. As M 613 has no P 9 -minor, and M 1 and M 2 are isomorphic to minors of M, we deduce 19

614 615 616 617 618 619 620 621 622 623 624 that neither si(m 1 ) nor si(m 2 ) has a P 9 -minor. By induction, the theorem holds for both si(m 1 ) and si(m 2 ). As M is not regular, at least one of si(m 1 ) and si(m 2 ), say si(m 1 ), is not regular. Claim: M 1 (and M 2 ) is simple unless both si(m 1 ) and si(m 2 ) are spikes. 625 Suppose not and we may assume that x in T has a parallel element x 1 in M 1. By Lemma 3.2, T is in a F 7 -minor of M 1 plus a parallel element x 1. By induction, si(m 2 ) is either regular and 3-connected, or one of the 16 internally 4-connected non-regular minors of Y 16 (thus is F 7 since it has a triangle); or is a spike or a starfish. Moreover, si(m 1 ) is either one of the 16 internally 4-connected non-regular minors of Y 16 (thus is F 7 ); or is a spike or a starfish. Suppose that si(m 2 ) is not a spike. Then either si(m 2 ) is regular or is a starfish. By Lemmas 3.10 and 3.16, either M 2 = M(K 4 ) where M(K 4 ) is 626 obtained from M(K 4 ) (which contains T ) by adding an element parallel to 627 an element of T, or T is in a rooted M(K 4 )-minor of M 2 using T (obtained 628 from M(K 4 ) containing T by adding an element parallel to either y or z). 629 In either case, as M is simple, we conclude that M contains a P 9 -minor, a 630 contradiction. Hence si(m 2 ) is a spike thus contains an F 7 -minor containing 631 T. Now if si(m 1 ) is not a spike, then si(m 1 ) is a starfish. Again using Lemma 632 3.10, it is easily checked that M has a P 9 -minor; a contradiction. Therefore 633 M 1 is simple unless both si(m 1 ) and si(m 2 ) are spikes. A similar argument 634 shows that M 2 is also simple unless both si(m 1 ) and si(m 2 ) are spikes. 635 636 637 638 639 640 Case 1: si(m 2 ) is regular. By Lemma 3.16, M 2 is either graphic or cographic. Moreover, (i) if M 2 is graphic, then either M 2 = M(G) where G is W4 or the Prism, or M 2 = M(K 4 ) where M(K 4 ) is obtained from M(K 4) (which contains T ) by adding an element parallel to an element of T ; and (ii) if M 2 is cographic but not graphic, then M = M (G), where G = K 3,n, 641 K 3,n, K 3,n, or K 3,n for some n 3. By the above claim, both M 1 and M 2 are simple. Moreover, M 1 is 3-642 643 644 645 646 647 648 649 650 651 652 653 connected, non-regular, and P 9 -free. By induction, M 1 is either one of the 16 internally 4-connected non-regular minors of Y 16 (therefore is F 7 as M 1 has a triangle); or M 1 is a spike or a starfish. That is, either M 1 is a spike or a starfish. If M 1 is a starfish, by Lemma 3.9, M = M 1 3 M 2 is also a starfish. Thus we may assume that M 1 is a spike which contains a triangle. Then M 1 is either F 7, S 8, Z s (s 4) or Z s \y s for some s 5. Suppose that M 1 is F 7. Then M = F 7 3 M 2 is either S 8 (not possible as M has at least 10 elements) or a starfish by the definition of a starfish. Suppose that M 1 is Z s (s 4) or Z s \y s for some s 5 and suppose that M 2 is not isomorphic to M(K 4 ). Then M 1 has a Z 4 -restriction containing T. Clearly, such restriction contains a F 7 -minor which is obtained from F 7 (which contains T ) by adding 20

664 654 an element parallel to the tip of the spike, say x in T. By Lemma 3.10, 655 T is in a M(K 4 )- minor of M 2 which is obtained from K 4 containing T by 656 adding an element parallel to an element z x of T. Thus we can find a 657 P 9 -minor in M, a contradiction. Suppose that M 1 is Z s (s 4) or Z s \y s for some s 5 and suppose that M 2 = M(K 658 4 ). If the extra element e of 659 M(K 4 ) added to M(K 4) is not parallel to x in M 2, then using the previously 660 mentioned F 7 -minor of M 1 containing T and the M(K 4 )-minor containing e, 661 we obtain a P 9 -minor of M; a contradiction. Now it is straightforward to see that M 662 = Z s+2 \y s+2 (s 4) which is a spike, or Z s+2 \y s, y s+2 (s 5). The 663 latter case does not happen as {y s, y s+2 } would be a 2-element cocircuit, but M is 3-connected. Finally we assume that M 1 = S8 = F 7 3 M(K 4 ) with tip 665 x. Then M = (F 7 3 M(K 4 3 M 2. By Lemma 3.4, M = F 7 3 (M(K 4 ) 3 666 M 2 ). By Corollary 3.16, M 2 is isomorphic to a 3-connected cographic matroid M (K 3,n ), M (K 3,n ), M (K 3,n (K 3,n 2 = M(K 4 ). If M 2 = M(K 668 4 ), then E(M) = 9; a contradiction. Thus M 2 is not isomorphic to M(K 4 ). By Corollary 3.8, M(K 4 ) 3 M 2 = M (G), where G = K 3,n, K 3,n, 670 or K 3,n 4 ) 3 M 2 contains a 2-element cocircuit which 671 does not meet any triangle of M(K 4 ) 3 M 2. In this case, by Corollary 3.6, 672 this 2-element cocircut would also be a cocircuit of M. As M is 3-connected, 673 we conclude that the latter does not happen, and that M is still a starfish. 674 675 676 677 678 679 Case 2: Neither M 1 nor M 2 is regular. By induction and the fact that both M 1 and M 2 have a triangle, that si(m 1 ) is either a spike containing a triangle or a starfish, and so is si(m 2 ). Case 2.1: Both si(m 1 ) and si(m 2 ) are starfishes. By the above claim, both M 1 and M 2 must be simple matroids. Now by Lemma 3.9, M is also a starfish. 689 680 Case 2.2: One of si(m 1 ) and si(m 2 ), say the former, is a spike. Suppose 681 that si(m 2 ) is a starfish. By the claim, both M 1 and M 2 are simple. As M 1 682 contains the triangle T, it is either Z s (s 3) or Z s \y s for some s 4. If M 1 = Z 3 = F7, by the definition of a starfish, M is also a starfish. If M 1 683 = Zs (s 4) 684 or Z s \y s for some s 5, then M 1 contains a Z 4 as a restriction which contains 685 T. But Z 4 contains a F 7 -minor containing T where F 7 is obtained from F 7 by 686 adding an element in parallel to the tip x of M 1. By Lemma 3.10, T is in a 687 M(K 4 )-minor of M 2 which is obtained from M(K 4 ) containing T by adding 688 an element parallel to y or z. We conclude that M contains a P 9 -minor, a contradiction. Now suppose that M 1 = Z4 \y 4 = S8 = F 7 3 M(K 4 ) with tip x. 690 Then M = (F 7 3 M(K 4 3 M 2. By Lemma 3.4, M = F 7 3 (M(K 4 ) 3 M 2 ). 691 By Corollary 3.9, M(K 4 ) 3 M 2 is either a starfish, or M(K 4 ) 3 M 2 and thus 692 M contains a 2-element cocircuit. As M is 3-connected, we conclude that the 693 latter does not happen, and that M is still a starfish by the definition of a 694 starfish. 21